SUMMARY: This chapter focuses on the properties and theorems related to circles, including tangents and their properties. KEY TOPICS: tangent to a circle, number of tangents from a point, theorems on tangents, tangent-segment theorem, secant-tangent theorem, angle subtended by a chord, cyclic quadrilaterals, properties of chords, arc and sector of a circle.
What is the radius of a circle if its diameter is 14 cm?
A7 cm
B14 cm
C21 cm
D28 cm
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Correct answer: Option 1 — 7 cm
Q21 Mark
If a circle has a circumference of 31.4 cm, what is its radius? (Use π = 3.14)
A5 cm
B10 cm
C15 cm
D20 cm
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Correct answer: Option 1 — 5 cm
Q31 Mark
Two tangents are drawn from a point outside a circle to the circle. If the lengths of the tangents are equal, what can be concluded about the point?
AIt lies on the circle
BIt is the center of the circle
CIt is equidistant from the center
DIt is outside the circle
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Correct answer: Option 3 — It is equidistant from the center
Q41 Mark
In a circle, if the angle subtended by an arc at the center is 80 degrees, what is the angle subtended by the same arc at any point on the remaining part of the circle?
A40 degrees
B80 degrees
C60 degrees
D20 degrees
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Correct answer: Option 1 — 40 degrees
Q51 Mark
A chord of a circle is 12 cm long and is 5 cm away from the center. What is the radius of the circle?
A10 cm
B13 cm
C15 cm
D20 cm
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Correct answer: Option 2 — 13 cm
Short Answer Questions5 questions
Q63 Marks
What is the definition of a circle?
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A circle is a set of all points in a plane that are at a fixed distance, called the radius, from a fixed point known as the center.
Q73 Marks
State the relationship between the radius and diameter of a circle.
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The diameter of a circle is twice the length of the radius. It is the longest chord of the circle and passes through the center.
Q83 Marks
How do you find the circumference of a circle?
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The circumference of a circle can be calculated using the formula C = 2πr, where r is the radius of the circle. Alternatively, it can also be expressed as C = πd, where d is the diameter.
Q93 Marks
What is the area of a circle and how is it calculated?
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The area of a circle is the space enclosed within its circumference. It is calculated using the formula A = πr², where r is the radius of the circle.
Q103 Marks
Explain the concept of a tangent to a circle.
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A tangent to a circle is a straight line that touches the circle at exactly one point. This point is called the point of tangency, and the tangent is perpendicular to the radius drawn to that point.
Long Answer Questions5 questions
Q116 Marks
Prove that the angle subtended by a chord at the center of a circle is twice the angle subtended by it at any point on the remaining part of the circle.
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Let O be the center of the circle and AB be a chord of the circle. Let C be any point on the circle such that the angle ACB is formed. The angle AOB, which is subtended by the chord AB at the center, is twice the angle ACB. This can be proved using the properties of isosceles triangles and the inscribed angle theorem. By constructing the triangles and analyzing the angles, we find that the relationship holds true, confirming the theorem.
Q126 Marks
A tangent is drawn to a circle from a point outside the circle. Prove that the length of the tangent from the external point to the point of tangency is equal to the square root of the difference between the square of the distance from the external point to the center of the circle and the square of the radius of the circle.
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Let O be the center of the circle, P be the external point, and T be the point of tangency. By applying the Pythagorean theorem to triangle OPT, we have PT² = OP² - OT², where OT is the radius of the circle. This shows that the length of the tangent PT can be expressed as PT = √(OP² - OT²), thus proving the required relationship between the lengths.
Q136 Marks
In a circle of radius 10 cm, two chords AB and CD intersect at point E. If AE = 4 cm, EB = 6 cm, and CE = 5 cm, calculate the length of the chord CD.
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Using the intersecting chords theorem, which states that AE × EB = CE × ED, we can substitute the known values. Here, AE = 4 cm, EB = 6 cm, and CE = 5 cm. Let ED be the unknown length of the segment. Thus, we have 4 × 6 = 5 × ED, leading to 24 = 5 × ED. Solving for ED gives ED = 24/5 = 4.8 cm. Therefore, the total length of chord CD is CE + ED = 5 cm + 4.8 cm = 9.8 cm.
Q146 Marks
A circle is inscribed in a triangle ABC. Prove that the sum of the distances from the incenter to the sides of the triangle is equal to the radius of the incircle multiplied by the semiperimeter of the triangle.
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Let the incenter be I and the radius of the incircle be r. The distances from I to the sides of triangle ABC can be represented as d1, d2, and d3. By the property of the incircle, the area of triangle ABC can be expressed in two ways: as the product of the semiperimeter s and the radius r (Area = s × r) and as the sum of the areas of the smaller triangles formed by the incenter and the sides. By equating these two expressions and simplifying, we arrive at the conclusion that the sum of the distances from the incenter to the sides equals r × s.
Q156 Marks
If two circles intersect at points A and B, prove that the line joining the centers of the circles bisects the angle formed by the tangents at points A and B.
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Let the centers of the two circles be O1 and O2. The tangents at points A and B create angles with the line segment O1O2. By using the properties of cyclic quadrilaterals and the angles subtended by the tangents, we can show that the angles formed are equal. By constructing the necessary triangles and applying the angle bisector theorem, we can conclude that the line joining the centers indeed bisects the angle formed by the tangents at points A and B, confirming the theorem.
Assertion–Reason Questions5 questions
Q161 Mark
Assertion (A): The radius of a circle is always half of its diameter.
Reason (R): The diameter is defined as twice the radius of a circle.
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Correct answer: Option 1 —
Both A and R are true, and R is the correct explanation of A.
Q171 Mark
Assertion (A): A tangent to a circle is perpendicular to the radius at the point of contact.
Reason (R): This is a fundamental property of tangents to circles.
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Correct answer: Option 1 —
Both A and R are true, and R is the correct explanation of A.
Q181 Mark
Assertion (A): The area of a circle is calculated using the formula A = πr^2.
Reason (R): The formula for the circumference of a circle is C = 2πr.
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Correct answer: Option 3 —
A is true, but R is false.
Q191 Mark
Assertion (A): Two circles can intersect at most at two points.
Reason (R): This is a property of the intersection of two circles in a plane.
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Correct answer: Option 1 —
Both A and R are true, and R is the correct explanation of A.
Q201 Mark
Assertion (A): The chord of a circle is always longer than the radius.
Reason (R): A chord can be equal to the radius only if it is a diameter.
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Correct answer: Option 3 —
A is true, but R is false.
Statement-Based Questions5 questions
Q211 Mark
Statement 1: The radius of a circle is always greater than its diameter.
Statement 2: The area of a circle is calculated using the formula A = πr².
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Correct answer: Option 4 —
Both statements are false.
Q221 Mark
Statement 1: A tangent to a circle is perpendicular to the radius at the point of contact.
Statement 2: The length of a tangent drawn from an external point to a circle is equal to the radius.
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Correct answer: Option 1 —
Both statements are true.
Q231 Mark
Statement 1: The circumference of a circle can be found using the formula C = 2πr.
Statement 2: All diameters of a circle are equal in length.
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Correct answer: Option 1 —
Both statements are true.
Q241 Mark
Statement 1: Two circles can intersect at more than two points.
Statement 2: The angle subtended by an arc at the center is double the angle subtended at any point on the remaining part of the circle.
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Correct answer: Option 2 —
Only Statement 1 is true.
Q251 Mark
Statement 1: The chord of a circle is always shorter than the diameter.
Statement 2: The center of a circle lies on its circumference.
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Correct answer: Option 4 —
Both statements are false.
